In 1954-1956 John Nash solved the isometric embedding problem for Riemannian Manifolds. Some of his results were extended to manifolds endowed with an indefinite metric tensor by Greene and Gromov/Rokhlin independently in 1970. Our goal is to extend these results to polyhedra. In the first half of the talk we will discuss results due to Zalgaller, Krat, Brehm, and Akopyan about piecewise linear (abbreviated PL) path isometries from Euclidean polyhedra into Euclidean space. We will then
use general position arguments, as well as tricks due to Nash and Petrunin, to show how these results can be extended to PL (and general) path isometric embeddings. In the second half of the talk we will generalize the definition of a Euclidean polyhedron to what is called a metric polyhedron. Results about simplicial and Pl isometric embeddings of metric polyhedra into Minkowski space (of an appropriate signature)
will be discussed. If time permits we will present the proofs of these
results.